Let \(y=x^{1-\delta _k+4\varepsilon }\) with \(\delta _3=\frac{2}{15},\,\delta _k=\frac{1}{k(2^{k-2}+1)}\) for \(4\leqslant k\leqslant 7\), and \(\delta _k=\frac{1}{k(k^2-k+1)}\) for \(k\geqslant 8\). In this paper, we establish an asymptotic formula of \(\mathcal {S}_k(x,y)\) and prove that